# The space-time fractional diffusion equation with Caputo derivatives

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## Abstract

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

## AMS Mathematics Subject Classification

26A33 49K20 44A10## Key words and phrases

Fractional diffusion equation time-space Caputo derivative Fourier transform Laplace transform Mittag-Leffler function Green function stable probability distributions## References

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